1. Edmunds-Karp Algorithm: Choosing Good Augmenting Paths
Use care when selecting augmenting paths.
Some choices lead to exponential algorithms.
Clever choices lead to polynomial algorithms.
Goal: choose augmenting paths so that:
Can find augmenting paths efficiently.
Few iterations.
Edmonds-Karp (1972): choose augmenting path with
Max bottleneck capacity. (fat path)
Sufficiently large capacity. (capacity-scaling)
Fewest number of arcs. (shortest path)

2. Shortest Augmenting Path
Intuition: choosing path via breadth first search.
Easy to implement.
may implement by coincidence!
Finds augmenting path with fewest number of arcs.
FOREACH e  E
f(e)  0
Gf  residual graph
WHILE (there exists augmenting path)
find such a path P by BFS
f  augment(f, P)
update Gf
RETURN f
ShortestAugmentingPath(V, E, s, t)

3. Shortest Augmenting Path: Overview of Analysis
L1. Throughout the algorithm, the length of the shortest path never decreases.
Proof ahead.
L2. After at most m shortest path augmentations, the length of the shortest augmenting path strictly increases.
Theorem. The shortest augmenting path algorithm runs in O(m2n) time.
O(m+n) time to find shortest augmenting path via BFS.
O(m) augmentations for paths of exactly k arcs.
If there is an augmenting path, there is a simple one.  1  k < n  O(mn) augmentations.

4. Shortest Augmenting Path: Analysis
Level graph of (V, E, s).
For each vertex v, define (v) to be the length (number of arcs) of shortest path from s to v.
LG = (V, EG) is subgraph of G that contains only those arcs (v,w)  E with (w) = (v) + 1. 2 5 G: s 3 6 t 2 5
LG: s 3 6 t
 = 0
 = 1
 = 2
 = 3

5. Shortest Augmenting Path: Analysis
Level graph of (V, E, s).
For each vertex v, define (v) to be the length (number of arcs) of shortest path from s to v.
L = (V, F) is subgraph of G that contains only those arcs (v,w)  E with (w) = (v) + 1.
Compute in O(m+n) time using BFS, deleting back and side arcs.
P is a shortest s-v path in G if and only if it is an s-v path L. 2 5 L: s 3 6 t
 = 0
 = 1
 = 2
 = 3

6. Shortest Augmenting Path: Analysis
L1. Throughout the algorithm, the length of the shortest path never decreases.
Let f and f' be flow before and after a shortest path augmentation.
Let L and L' be level graphs of Gf and Gf '
Only back arcs added to Gf '
path with back arc has length greater than previous length 2 5 L s 3 6 t
 = 0
 = 1
 = 2
 = 3 2 5 L' s 3 6 t

7. Shortest Augmenting Path: Analysis
L2. After at most m shortest path augmentations, the length of the shortest augmenting path strictly increases.
At least one arc (the bottleneck arc) is deleted from L after each augmentation.
No new arcs added to L until length of shortest path strictly increases. 2 5 L s 3 6 t
 = 0
 = 1
 = 2
 = 3 2 5 L' s 3 6 t

8. Shortest Augmenting Path: Review of Analysis
L1. Throughout the algorithm, the length of the shortest path never decreases.
L2. After at most m shortest path augmentations, the length of the shortest augmenting path strictly increases.
Theorem. The shortest augmenting path algorithm runs in O(m2n) time.
O(m+n) time to find shortest augmenting path via BFS.
O(m) augmentations for paths of exactly k arcs.
O(mn) augmentations.
Note: (mn) augmentations necessary on some networks.

9. m3/2 log (n2 / m) log U mn2/3 log (n2 / m) log U
History
Year
Discoverer
Method
Big-Oh
1951
Dantzig
Simplex
mn2U
1955
Ford, Fulkerson
Augmenting path
mnU
1970
Edmonds-Karp
Shortest path
m2n
1970
Dinitz
Shortest path
mn2
1972
Edmonds-Karp, Dinitz
Capacity scaling
m2 log U
1973
Dinitz-Gabow
Capacity scaling
mn log U
1974
Karzanov
Preflow-push n3
1983
Sleator-Tarjan
Dynamic trees
mn log n
1986
Goldberg-Tarjan
FIFO preflow-push
mn log (n2 / m)
. . .
. . .
. . .
. . .
1997
Goldberg-Rao
Length function
m3/2 log (n2 / m) log U mn2/3 log (n2 / m) log U